Every hour the relative distance between the friends reduces and hence the bird need not fly the whole 36 miles everytime. The first time when the bird travels 27 miles it wld have gone to the cyclist who would have covered 9 miles from the other end, this way everytime the cyclist come closer to each other and the bird travels less distance in every iteration. given answer of 54 miles is wrong.
the reason 54 is right, is because the boys are traveling a total of 12 MPH seeing how they are traveling towards each other their speeds can be added. Giving you the equation of 12t=36 simplifying this gives you t=3 and the bird is flying at 18 MPH meaning that it will fly 18t for t hours, meaning that they travelled 3 hours it will fly for 3 hours so 18 x 3 = 54
The first time i put in 54miles and it said incorrect so i did the problem again still i got 54miles so i looked at the anwser and it only said 54.......Im so mad
The bird will be traveling at 18 mph for the same amount of time that the cyclists are traveling. Since the riders are traveling at 6 mph and are meeting at a point 18 miles from their respective starting points, the total time of travel comes to 3 hours. And 3 times 18 is 54; exactly the given answer. You will also notice, CR, that your method of finding the solution will also yield the correct answer if carried out properly. The separation of the two cyclists in a graphical sense is given by the line y= -12x+36. Integrating this line on the interval from zero to 3 with respect to x also gives the solution 54. This is the limit that CR aluded to earlier.
if the bird is traveling between the two friends, and the bird starts flying the instant the two start traveling, the bird would not fly the entire distance (from person A's start point to person B's start point), because person B would have traveled some distance by the time the bird reaches him.
spideys got it. everyone else is way overthinking it. the bird is flying at a constant velocity, and the time the friends are riding is the same as the bird, so for the bird, the distance of the friends is irrelevant. half way between 36 miles is 18 miles. at 6 mph it takes 3 hours to cover 18 miles. since the bird flies the same amount of time as the friends, 3 hours, he'll have traveled 54 miles(18 miles*3 hours=54 miles)
The first time I did this puzzle it was hard but know that I think of it it is easy/I think that we could use this in the futher as a fun game for kids to do in there spare time and it could mame many people think really hard,Over all I am very pleased with this problem,When I get home I am going to show my mom this problem when I get home after a while
John Von Neumann ,famous Math Genius, was supposedly asked this question and he gave the answer in a few seconds. The person who asked him remarked that "wasn't that a neat trick" Von Neumann replied, "I just summed the infinite series."
It works this way too... In the first 1.5 hours, the bikers each travel 9m (putting them at 9m and 27m respectively); in 1.5 hours, the bird will also travel 27; meaning at 1.5 hours (which happens to be half the time), the bird meets the 2nd rider (27m).
It turns around and heads back to the first guy .75 hours later, the bird travels 13.5m back (back to the point 13.5m). In that .75 hours, the biker travels 4.5m from 9m to 13.5m, meaning the bird meets the biker again at that point.
You can see continuing that the bird's trips will take half as long and be half the distance every time it reaches a biker. Adding, we can see that:
etc. Eventually the theoretical bird is bouncing back and forth making a one inch trip, a half inch trip, a quarter inch trip, etc. That's why it's a theoretical question. The solution is the limit of the sum of:
27/2^n for n starting from 0 and approaching infinity, which as you can see from the above, basic addition, is 54, which is 1.5 times the distance the two bikers cover (at 12mph combined, 1.5 times slower than the bird)
or is there something i missed?
Thanks for the fun problems
where on earth did u learn that?!?
It works this way too... In the first 1.5 hours, the bikers each travel 9m (putting them at 9m and 27m respectively); in 1.5 hours, the bird will also travel 27; meaning at 1.5 hours (which happens to be half the time), the bird meets the 2nd rider (27m).
It turns around and heads back to the first guy .75 hours later, the bird travels 13.5m back (back to the point 13.5m). In that .75 hours, the biker travels 4.5m from 9m to 13.5m, meaning the bird meets the biker again at that point.
You can see continuing that the bird's trips will take half as long and be half the distance every time it reaches a biker. Adding, we can see that:
27 + 13.5 = 40.5
+ 6.75 = 47.25
+ 3.375 = 50.625
+ 1.6875 = 52.3125
+ 0.84375 = 53.15625
+ 0.421875 = 53.578125
+ 0.2109375 = 53.7890625
+ 0.10546875 = 53.89453125
+ 0.052734375 = 53.947265625
etc. Eventually the theoretical bird is bouncing back and forth making a one inch trip, a half inch trip, a quarter inch trip, etc. That's why it's a theoretical question. The solution is the limit of the sum of:
27/2^n for n starting from 0 and approaching infinity, which as you can see from the above, basic addition, is 54, which is 1.5 times the distance the two bikers cover (at 12mph combined, 1.5 times slower than the bird)